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Nash blowup fails to resolve singularities in dimensions four and higher

Federico Castillo, Daniel Duarte, Maximiliano Leyton-Álvarez, Alvaro Liendo

TL;DR

The paper disproves the Nash blowup conjecture and the normalized Nash blowup conjecture in dimensions at least $4$ over any algebraically closed field. It constructs explicit normal affine toric varieties in dimension $4$ (and analyzes higher dimensions) that persist as charts isomorphic to the original variety within iterations of the Nash blowup or normalized Nash blowup, across characteristics, using a combinatorial toric framework based on Hilbert bases, determinants modulo $p$, and saturations. The main results are realized through careful toric constructions: in characteristic $0$ a chart $X(S_A)$ with $A$ chosen from the Hilbert basis yields $X(S_A)\,\cong\,X(S)$; in characteristic $2$ and $3$ analogous counterexamples are exhibited, with the latter requiring two iterations to realize the phenomenon. Complementary computer experiments in SageMath corroborate and illuminate the constructions, revealing intricate looping behavior and illustrating termination phenomena in higher dimensions. Overall, the work shows that iterative Nash-type procedures cannot provide a canonical resolution algorithm beyond dimension $3$, even after normalization, and clarifies characteristic-dependent behavior in toric settings.

Abstract

In this paper we show that iterating Nash blowups or normalized Nash blowups does not resolve the singularities of algebraic varieties of dimension four or higher over an algebraically closed field of arbitrary characteristic.

Nash blowup fails to resolve singularities in dimensions four and higher

TL;DR

The paper disproves the Nash blowup conjecture and the normalized Nash blowup conjecture in dimensions at least over any algebraically closed field. It constructs explicit normal affine toric varieties in dimension (and analyzes higher dimensions) that persist as charts isomorphic to the original variety within iterations of the Nash blowup or normalized Nash blowup, across characteristics, using a combinatorial toric framework based on Hilbert bases, determinants modulo , and saturations. The main results are realized through careful toric constructions: in characteristic a chart with chosen from the Hilbert basis yields ; in characteristic and analogous counterexamples are exhibited, with the latter requiring two iterations to realize the phenomenon. Complementary computer experiments in SageMath corroborate and illuminate the constructions, revealing intricate looping behavior and illustrating termination phenomena in higher dimensions. Overall, the work shows that iterative Nash-type procedures cannot provide a canonical resolution algorithm beyond dimension , even after normalization, and clarifies characteristic-dependent behavior in toric settings.

Abstract

In this paper we show that iterating Nash blowups or normalized Nash blowups does not resolve the singularities of algebraic varieties of dimension four or higher over an algebraically closed field of arbitrary characteristic.
Paper Structure (3 sections, 3 theorems, 34 equations, 1 figure)

This paper contains 3 sections, 3 theorems, 34 equations, 1 figure.

Table of Contents

  1. The Nash blowup of an affine toric variety
  2. Proof of the Theorem
  3. Remarks on computer experimentation

Key Result

Theorem 1

For every $d\geq 4$ and every algebraically closed field $\mathbf{k}$, there exists a normal singular affine algebraic variety $X$ of dimension $d$ over $\mathbf{k}$ such that: Consequently, in every dimension $d \geq 4$ and every characteristic, iterating the Nash blowup or the normalized Nash blowup fails to resolve singularities.

Figures (1)

  • Figure 1: The combinatorics of an affine slice of $\omega$. The vector $h_7$ lies on the face $\operatorname{Cone}(h_2,h_4,h_5)$.

Theorems & Definitions (6)

  • Theorem
  • Lemma 1
  • Lemma 2
  • proof
  • Remark 3
  • Remark 4